Proof
Foundations and logic
A finite chain of deductive steps, each justified by an axiom or an already established result, that establishes a statement within a formal system.
What is usually left out
A proof is checkable in principle by following its steps, which is what separates it from evidence. Numerical verification, however extensive, is not a proof: checking a claim for every case up to 10²⁰ says nothing about 10²⁰ + 1 unless an argument covers all cases at once.